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A329864 Number of compositions of n with the same runs-resistance as cuts-resistance. 7
1, 0, 0, 0, 0, 2, 5, 10, 17, 27, 68, 107, 217, 420, 884, 1761, 3679, 7469, 15437, 31396, 64369 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

A composition of n is a finite sequence of positive integers summing to n.

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined to be the number of applications required to reach a singleton.

For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.

LINKS

Table of n, a(n) for n=0..20.

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

EXAMPLE

The a(5) = 2 through a(8) = 17 compositions:

  (1112)  (1113)   (1114)    (1115)

  (2111)  (1122)   (1222)    (1133)

          (2211)   (2221)    (3311)

          (3111)   (4111)    (5111)

          (11211)  (11122)   (11222)

                   (11311)   (11411)

                   (21112)   (12221)

                   (22111)   (21113)

                   (111121)  (22211)

                   (121111)  (31112)

                             (111131)

                             (111221)

                             (112112)

                             (112211)

                             (122111)

                             (131111)

                             (211211)

For example, the runs-resistance of (111221) is 3 because we have: (111221) -> (321) -> (111) -> (3), while the cuts-resistance is also 3 because we have: (111221) -> (112) -> (1) -> (), so (111221) is counted under a(8).

MATHEMATICA

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&, q, Length[#]>1&]]-1;

degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&, q, Length[#]>0&]]-1;

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], runsres[#]==degdep[#]&]], {n, 0, 10}]

CROSSREFS

The version for binary expansion is A329865.

Compositions counted by runs-resistance are A329744.

Compositions counted by cuts-resistance are A329861.

Compositions with runs-resistance = cuts-resistance minus 1 are A329869.

Cf. A003242, A098504, A114901, A242882, A318928, A319411, A319416, A319420, A319421, A329867, A329868.

Sequence in context: A062493 A056871 A246883 * A174910 A301273 A007504

Adjacent sequences:  A329861 A329862 A329863 * A329865 A329866 A329867

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Nov 23 2019

STATUS

approved

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Last modified April 15 01:24 EDT 2021. Contains 342974 sequences. (Running on oeis4.)